Teaching Geometry: Reader question – which proofs?

I’m a math teacher in NYC and I have been reading your excellent blog for the past couple of years. I find I agree with many of your opinions on math content – in particular things that are important and things that aren’t.

Without boring you with details, I’m involved in planning the Geometry curriculum for my school this summer and I am asking several people I know about what they think are the top 10 (20?) proofs for high school Geometry. I’m of the Michael Serra camp that believes that it’s far more important to be able to prove general, important theorems (sum of interior angles in a triangle, measurement of an inscribed angle in a circle) from scratch than to prove two specific triangles congruent once given 75% of the information you need.

If you have time to respond (via email or perhaps as a post on your blog – I know you’re busy) about your list of top proofs, I would love to hear from you. It’s not clear to me whether you are actually teaching the new Geometry curriculum in your school, or if you’re simply watching the Geometry regents as a concerned observer, but any input you have would be appreciated.

Thanks for your time and keep up the great blogging!

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Last question first. Last year I taught the first term of geometry to students who will complete the course this Fall. I taught a fairly traditional course, with constructions (straight edge and compass) woven in from the beginning. These posts describe a theorem informing construction in one incident and construction informing a theorem in another in that class.

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There are choices with proof:

The two specific triangles proofs (fairly standard. Look at this diagram. Prove something specific about the given situation)

Required proofs. Here’s a list of 8 or 24 or 47 proofs that you must be able to rattle off. Good luck studying/memorizing!

The first case, as Serra points out, is fairly artificial. And, it’s not what we do when we do math later on.

The second case is usually rote. Look, memorizing proofs has value, you learn how things fit together. But there is no sense of creation in repeating a proof you’ve read.

You could also ask them for an important proof without telling them in advance. But I’m not sure what you’d be testing.

You could also ask them to prove a known theorem that they’ve just worked on. There is a huge danger there, which is, the kid will produce a proof that relies on corollaries of the thing they are proving. It is very easy to fall into circular reasoning (begging the question). I learned this the hard way (sorry Ryan, Hyatt, Alec, Mychaela, Tova, Veronica, Hiroshi, Valerie, Yenyu, Afsana, Mike, Sara, Sangeetha, Stephanie….)

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I have become attached to a third option. Ask the kids to prove things that look like theorems or corollaries. Let them be general (not category 1, above). Let them not be real theorems, or at least not major enough to be mentioned in a book. And let the kids prove. There is creation. It is not a test of rote memorization. No other theorem depends upon this, so there is no danger of circular reasoning.

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Prove, in a trapezoid the altitude is perpendicular to both bases.

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Prove, the figure formed by joining the consecutive midpoints of a rectangle is a rhombus.

(this is a fairly standard coordinate proof, but can be used as a novel proof before coordinates are introduced. In fact, a significant number of standard coordinate proofs can be used here, eg, prove that the figure formed by joining consecutive midpoints of an arbitrary quadrilateral form a parallelogram.)

I think that when we ask students to write proofs in Geometry we are hopefully giving them a chance to:

a. learn a concept that will stay with them after the class

and

b. give them a taste for what deductive proof is like in mathematics.

To me, the proofs of type 1. in your post do neither of these things. I agree that they’re not great. The proofs of type 2. ideally do this, but your criticisms about rote learning are legitimate.

On my final last year, I asked students to prove, among other things, these two theorems (in order, as the second is dependent on the first):

1. The sum of the interior angles of a triangle is 180 degrees.

2. The measure of the exterior angle of a triangle is equal to the sum of the two remote interior angles.

Students had to make and label their own drawings, and then write a paragraph proof. I agree that there is a real danger of students simply memorizing proofs without thinking about what they’re writing, but I’ve found that having them write paragraph-style proofs (rather than two-column with abbreviations or (*shudder*) numbered theorems given as reasons when they may not remember what the abbreviations or numbers mean) helps avoid this.

I think your idea of having them prove seemingly trivial but original is a step in the right direction, but I think it also runs the risk of giving them the impression that proofs in geometry are for exhibiting facts that are readily obvious. I do like the fact that they get a sense of “creation” as you mentioned above, and obviously proving major theorems that have been around for hundreds of years lacks this.

I guess my original question was: what theorems or concepts from geometry do I want them to be able to explain at a cocktail party ten years from now? For example, we did an informal proof (no limits, no calculus obviously) of the formula for the area of the circle this year. My fellow teachers were shocked that I simply didn’t just feed them the formula (one even went so far as to say “It’s given on the SAT!”) Don’t we want students to know where these ideas come from?

You’re correct in saying that there’s little value in rote memorization (especially from a colossal list of 40 or 50 proofs – I’ve seen this done), but I have a hard time finding a compromise between the the two types of proof you list above. I appreciate your suggestion and it’s given me some things to think about.